Graph

Data Structures Linear data structures: –Array, linked list, stack, queue Non linear data structures: –Tree, binary tree, graph and digraph

Graph A graph is a non-linear structure (also called a network) Unlike a tree or binary tree, a graph does not have a root Any node (vertex, v) can be connected to any other node by an edge (e) Can have any number of edges (E) and nodes (set of Vertex V) Application: the highway system connecting cities on a map a graph data structure

Graphs G = (V,E) V is the vertex set (nodes set). Vertices are also called nodes and points. E is the edge set. Each edge connects two different vertices. Edges are also called arcs and lines. Directed edge has an orientation (u,v). uv

Graphs Undirected edge has no orientation (u,v). uv Undirected graph => no oriented edge. Directed graph => every edge has an orientation. uv

Undirected Graph G = (V, E) V = {1,2,…11} E = { (2,1), (4,1), …}

Directed Graph (Digraph) G = (V, E) V = {1,2,…11} E = { (1,2), (1,4), …}

Applications—Communication Network Vertex = city, edge = communication link

Driving Distance/Time Map Vertex = city, edge weight = driving distance/time. Find a shortest path from city 1 to city 5 –1  2  5 (12) –1  4  6  7  5 (14)

Street Map Some streets are one way. Find a path from 1 

Complete Undirected Graph Has all possible edges. n = 1 n = 2 n = 3 n = 4

Number Of Edges—Undirected Graph Each edge is of the form (u,v), u != v. Number of such pairs in an n vertex graph is n(n-1). Since edge (u,v) is the same as edge (v,u), the number of edges in a complete undirected graph is n(n-1)/2. Number of edges in an undirected graph is <= n(n-1)/2.

Complete directed Graph Has all possible edges. n = 1 n = 2 n = 3 n = 4

Number Of Edges—Directed Graph Each edge is of the form (u,v), u != v. Number of such pairs in an n vertex graph is n(n-1). Since edge (u,v) is not the same as edge (v,u), the number of edges in a complete directed graph is n(n-1). Number of edges in a directed graph is <= n(n-1).

Vertex Degree Number of edges incident to vertex. degree(2) = 2, degree(5) = 3, degree(3) =

Sum Of Vertex Degrees Sum of degrees = 2e (e is number of edges)

In-Degree Of A Vertex in-degree is number of incoming edges indegree(2) = 1, indegree(8) =

Out-Degree Of A Vertex out-degree is number of outbound edges outdegree(2) = 1, outdegree(8) =

Sum Of In- And Out-Degrees each edge contributes 1 to the in-degree of some vertex and 1 to the out-degree of some other vertex sum of in-degrees = sum of out-degrees = e, where e is the number of edges in the digraph

Graph Operations And Representation

Sample Graph Problems Path problems. Connectedness problems. Spanning tree problems.

Path Finding Path between 1 and Path length is 20.

Another Path Between 1 and Path length is 28.

Example Of No Path No path between 2 and

Connected Graph Undirected graph. There is a path between every pair of vertices.

Example Of Not Connected

Connected Graph Example

Connected Components

Connected Component A maximal subgraph that is connected. A connected graph has exactly 1 component.

Not A Component

Communication Network Each edge is a link that can be constructed

Communication Network Problems Is the network connected?  Can we communicate between every pair of cities? Find the components. Want to construct smallest number of links so that resulting network is connected (finding a minimum spanning tree).

Cycles And Connectedness Removal of an edge that is on a cycle does not affect connectedness.

Cycles And Connectedness Connected subgraph with all vertices and minimum number of edges has no cycles.

Tree Connected graph that has no cycles. n vertex connected graph with n-1 edges.

Graph Representation Adjacency Matrix Adjacency Lists  Linked Adjacency Lists  Array Adjacency Lists

Adjacency Matrix 0/1 n x n matrix, where n = # of vertices A(i,j) = 1 iff (i,j) is an edge

Adjacency Matrix Properties Diagonal entries are zero. Adjacency matrix of an undirected graph is symmetric.  A(i,j) = A(j,i) for all i and j.

Adjacency Matrix (Digraph) Diagonal entries are zero. Adjacency matrix of a digraph need not be symmetric.

Adjacency Matrix n 2 bits of space For an undirected graph, may store only lower or upper triangle (exclude diagonal).  (n-1)n/2 bits

Adjacency Lists Adjacency list for vertex i is a linear list of vertices adjacent from vertex i. An array of n adjacency lists aList[1] = (2,4) aList[2] = (1,5) aList[3] = (5) aList[4] = (5,1) aList[5] = (2,4,3)

Linked Adjacency Lists Each adjacency list is a chain aList[1] aList[5] [2] [3] [4] Array Length = n # of chain nodes = 2e (undirected graph) # of chain nodes = e (digraph)

Array Adjacency Lists Each adjacency list is an array list aList[1] aList[5] [2] [3] [4] Array Length = n # of list elements = 2e (undirected graph) # of list elements = e (digraph)

Number Of C++ Classes Needed Graph representations  Adjacency Matrix  Adjacency Lists  Linked Adjacency Lists  Array Adjacency Lists  3 representations Graph types  Directed and undirected.  Weighted and unweighted.  2 x 2 = 4 graph types 3 x 4 = 12 C++ classes

Abstract Class Graph template class graph { public: // ADT methods come here // implementation independent methods come here };

Abstract Methods Of Graph // ADT methods virtual ~graph() {} virtual int numberOfVertices() const = 0; virtual int numberOfEdges() const = 0; virtual bool existsEdge(int, int) const = 0; virtual void insertEdge(edge *) = 0; virtual void eraseEdge(int, int) = 0; virtual int degree(int) const = 0; virtual int inDegree(int) const = 0; virtual int outDegree(int) const = 0;

Abstract Methods Of Graph // ADT methods (continued) virtual bool directed() const = 0; virtual bool weighted() const = 0; virtual vertexIterator * iterator(int) = 0; virtual void output(ostream&) const = 0;